On the gcd graphs over polynomial rings

TL;DR

This paper extends the concept of gcd-graphs from integers modulo n to polynomial rings over finite fields, exploring their properties and spectral characteristics.

Contribution

It introduces and studies gcd-graphs over polynomial rings, establishing foundational properties and drawing analogies to classical gcd-graphs over integers.

Findings

Gcd-graphs over polynomial rings have integral spectra.

Fundamental properties of these graphs are established.

Analogies between number fields and function fields are demonstrated.

Abstract

Gcd-graphs over the ring of integers modulo $n$ are a natural generalization of unitary Cayley graphs. The study of these graphs has foundations in various mathematical fields, including number theory, ring theory, and representation theory. Using the theory of Ramanujan sums, it is known that these gcd-graphs have integral spectra; i.e., all their eigenvalues are integers. In this work, inspired by the analogy between number fields and function fields, we define and study gcd-graphs over polynomial rings with coefficients in finite fields. We establish some fundamental properties of these graphs, emphasizing their analogy to their counterparts over $\mathbb{Z}.$